Speak Maths · University

Fields: wind, whirl and flow

When every point of space carries an arrow, like the wind, new questions appear: what adds up over a patch, what flows through an edge, and does the air spread out or swirl?

Draft: written by an AI helper and not yet checked by a teacher.

The alphabet

Read each sign aloud, the way it is written under it.

$F (x, y) = (P, Q)$ Say: F of x and y equals P and QMeans: a vector field: an arrow at every point, like the wind with its own direction and strength everywhere
$\iint_{R} f d A$ Say: the double integral of f over R, d AMeans: add up f over every tiny piece of the flat region R; with f equal to 1 it is the area
$∭_{V} f d V$ Say: the triple integral of f over V, d VMeans: add up f over every tiny piece of the solid V; with f equal to 1 it is the volume
$\oint_{C} F \cdot d r$ Say: the line integral round C of F dot d rMeans: the total push of the field along one full trip round the closed path C: the circulation
$\nabla \cdot F$ Say: the divergence of F, or del dot FMeans: a number at each point: positive where the field spreads out, negative where it squeezes in
$\nabla \times F$ Say: the curl of F, or del cross FMeans: how much the field swirls at each point; an arrow along the axis of the swirl

The story: Wind over a field, water through a net

Subtitles:

Tap a line to see its words.

$F (x, y) = (- y, x)$ Wind swirls round the middle of a flat field. At the spot x metres east and y north, it blows (−y, x).
$F (1, 0) = (0, 1)$ One metre east of the middle, the wind blows due north at 1 metre per second.
$\nabla \cdot F = 0$ The air goes round and round, so it nowhere piles up or thins out: the divergence is zero.
$\nabla \times F = (0, 0, 2)$ But it swirls. A tiny paddle wheel would spin anticlockwise, with strength 2 about the upright axis.
$\iint_{R} 2 d A = 2 π$ Over a round patch R of radius 1, with area π, the swirl strength 2 adds up to 2π.
$\oint_{C} F \cdot d r = 2 π$ A walker once round the edge C of the patch, with the wind behind her all the way, feels the same total: 2π.
$G (x, y, z) = (x, y, z)$ Now water. A tap in the middle of a ball-shaped net sends water straight outwards: at each point it flows (x, y, z).
$\nabla \cdot G = 3$ Everywhere inside, the water is spreading out at the same rate: the divergence is 3.
$∭_{V} 3 d V = 4 π$ Over the ball V of radius 1, whose volume is four thirds of π, the spreading adds up to 4π: exactly what leaves through the net.

In words this story is 895 letters long; in symbols, 85.

Read it

What does it say?

$\iint_{R} 1 d A$

$\nabla \cdot F > 0$

$\nabla \times F = 0$

$\oint_{C} F \cdot d r$

$F (x, y) = (0, - 9.8)$

$∭_{V} 1 d V$

$\nabla \cdot F < 0$

$\nabla \times F = (0, 0, - 2)$

$\oint_{C} F \cdot d r = 0$

$\iint_{R} \nabla \times F d A$

$F (2, 3) = (1, 0)$

$\nabla \cdot F = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}$

Write it

Which notation says this?

The total of f over the flat region R.

The divergence of the field F.

The total push of F along one trip round the closed curve C.

The volume of the solid V.

The curl of the field F.

A field that at every point (x, y) blows due north with strength 3.

The field neither spreads out nor squeezes in anywhere.

The swirl of F added up over the flat patch R.

The total of f over the solid V.

The total push of F along one trip round the closed loop C is 6π.

The spreading of F added up over the whole solid V is 12.

A field with no swirl anywhere.

Build it

Tap the tiles in order. One tile does not belong.

The double integral of f over R, d A.

The triple integral over V of the divergence of G, d V, equals four pi.

The curl of F equals the zero vector.

The line integral round C of F dot d r equals two pi.

The field F of x and y equals minus y and x.

The triple integral over V of 1, d V, equals four thirds pi.

Challenges

Creative puzzles that use every sign of this chapter. Take your time; there is no clock.

1 Four winds, three tests

Four wind fields over a flat field, and three tests for each. The table was checked for you; use it, and the idea that going round a loop adds up the swirl inside, to judge each sentence.

	$\nabla \cdot F = 0$	$\nabla \times F = 0$	$\oint_{C} F \cdot d r = 0$
$F_{1} = (x, y)$		✓	✓
$F_{2} = (- y, x)$	✓
$F_{3} = (y, x)$	✓	✓	✓
$F_{4} = (x - y, x + y)$

A tick means the column’s equation holds for the row’s wind. C is the unit circle, once round anticlockwise. In the plane, ∇×F means the swirl ∂Q/∂x − ∂P/∂y.

$\nabla \cdot (x, y) = 2$
$\nabla \times (- y, x) = 2$
$\oint_{C} (y, x) \cdot d r = 0$
$\oint_{C} (x - y, x + y) \cdot d r = 2 π$
$\nabla \cdot (y, x) = 1$
$\nabla \times (x, y) = 2$

2 The field dictionary

Six signs for arrows, sums and flows. Pair each with its meaning.

Tap a line, then its meaning.

3 Spreading out

This should be the formula for how much a plane field spreads out. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

4 A number, not an arrow

Which answer is the divergence?

$F = (x^{2}, y^{2}, z^{2}), \nabla \cdot F = ?$

5 Tile the floor

Add up a constant 3 over the rectangle.

$\iint_{R} 3 d A, R = [0, 2] \times [0, 5]$

6 Once round the circle

A swirling wind blows round a circle of radius one. What does the walker feel in total?

$F = (- y, x), C : x^{2} + y^{2} = 1, \oint_{C} F \cdot d r = ?$

7 What leaves the ball

Water spreads from the middle of a ball V of radius 1 with ∇·F = 3 everywhere. Write in symbols that the total spreading in the ball is 4π. Then compare.

Your version

8 The swirl law

The total push round the edge of a flat patch equals the swirl added up over the patch. Write it in symbols. Then compare.

Your version

9 Steady wind, fountain, whirl, saddle

Four winds over a flat field: a steady breeze, a fountain, a whirl and a saddle. The table was checked for you. Judge each sentence; remember that going round the loop adds up the swirl inside.

	$\nabla \cdot F = 0$	$\nabla \times F = 0$	$\oint_{C} F \cdot d r = 0$
$F_{1} = (0, 3)$	✓	✓	✓
$F_{2} = (2 x, 2 y)$		✓	✓
$F_{3} = (- 2 y, 2 x)$	✓
$F_{4} = (x, - y)$	✓	✓	✓

A tick means the column’s equation holds for the row’s wind. C is the unit circle, once round anticlockwise. In the plane, ∇×F means ∂Q/∂x − ∂P/∂y.

$\nabla \cdot (2 x, 2 y) = 4$
$\nabla \times (- 2 y, 2 x) = 4$
$\oint_{C} (- 2 y, 2 x) \cdot d r = 4 π$
$\nabla \cdot (x, - y) = 2$
$\nabla \times (0, 3) = 3$
$\oint_{C} (2 x, 2 y) \cdot d r = 4 π$

10 Taps and drains

Water flows over a floor as F = (x², y). Four spots are listed; the table says at each whether water spreads out (a tap), whether it swirls, and whether the flow there is exactly (1, 0). Judge each sentence.

	$\nabla \cdot F > 0$	$\nabla \times F = 0$	$F = (1, 0)$
$(1, 0)$	✓	✓	✓
$(- 1, 0)$		✓	✓
$(0, 2)$	✓	✓
$(- 2, 1)$		✓

A tick means the column holds at the row’s point for F = (x², y). ∇×F is the plane swirl ∂Q/∂x − ∂P/∂y.

$\nabla \cdot F (1, 0) = 3$
$\nabla \cdot F (- 2, 1) = - 3$
$\nabla \times F = 0$
$F (- 1, 0) = (- 1, 0)$
$\nabla \cdot F (0, 2) < 0$
$F (- 2, 1) = (4, 1)$

11 Five totals

Five sums that look alike. Pair each with what it adds up.

Tap a line, then its meaning.

12 Swirling

This should be the formula for how much a plane field F = (P, Q) swirls. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

13 Ball of radius one

This should say: adding 1 over every tiny piece of the ball V of radius 1 gives its volume, four thirds of π. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

14 A steady wind round a circle

The wind blows due east at strength 3 everywhere. A walker goes once round a circle of radius 1. What is the total push along the way?

$F = (3, 0), C : x^{2} + y^{2} = 1, \oint_{C} F \cdot d r = ?$

15 Under a sloping roof

A roof of height x + y stands over a 2-by-2 square floor. How much air is under it?

$\iint_{R} (x + y) d A, R = [0, 2] \times [0, 2]$

16 The river that swirls

A river flows east, faster the further north you go: F = (y, 0). Does it swirl, and which way?

$F = (y, 0), \nabla \times F = ?$

17 Everything falls

Near the ground, gravity pulls every point straight down with the same strength g. Write this as a field, then compare.

Your version

Tell your own

Write a two-line story of a flow: smoke, a stream or crowds in a hall. Say where it spreads out or swirls, in signs only.

Kept on this device only.

Fields: wind, whirl and flow

The alphabet

The story: Wind over a field, water through a net

Read it

Write it

Build it

Challenges

1 Four winds, three tests

2 The field dictionary

3 Spreading out

4 A number, not an arrow

5 Tile the floor

6 Once round the circle

7 What leaves the ball

8 The swirl law

9 Steady wind, fountain, whirl, saddle

10 Taps and drains

11 Five totals

12 Swirling

13 Ball of radius one

14 A steady wind round a circle

15 Under a sloping roof

16 The river that swirls

17 Everything falls

Tell your own

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